Multi-level printing process reducing aliasing in graphics

ABSTRACT

A multi-level printing process using a printer by converting electronic page data to multi-level bitmap data with a specific number of gradation levels Nlev at a specified spatial resolution higher than that of the printer; converting the multi-level bitmap data to a lower spatial resolution; reducing the number of gradation levels by a processing technique to some lower number Nlevp giving a second multi-level bitmap; and printing the multi-level output bitmap by a printer having at least Nlevp different gradation levels at a single pixel location.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of under 35 U.S.C. §119(e) for U.S.Provisional Patent Application Ser. No. 60/245,736 filed on Nov. 3,2000.

FIELD OF THE INVENTION

This invention relates to systems and procedures for printing documents.These documents can contain all types of visual information, such astext, graphics, images, etc., further commonly referred to as “image”

BACKGROUND OF THE INVENTION

A well-known problem that is common to all recording systems, thatconvert image data into discretised pixel information, is the occurrenceof reconstruction errors, commonly called ‘aliasing’. It results injagged edges at lines or curves that are not aligned with the discretepixel grid of the recording apparatus or printer.

This problem can be cured in two ways. The first one is to increase thespatial resolution of the recording device. This is often a technicallydifficult and costly solution.

A second approach, and this is the approach to which the presentinvention relates, is to trade spatial resolution with tonal resolutionin order to improve the visual quality.

This approach is common for font rendering on computer displays. Hereblack pixels are surrounded by grey pixels. This results in visualsmoothing of the edges, rendering the edges in a way more pleasing tothe human eye.

A procedure is disclosed that creates multi-level gradation data from anabstract page description to achieve a similar result. Most documents tobe printed contain a mixture of text, graphics and images and are givenin the form of an page description. These documents may be PDFdocuments, PostScript™ files, MS-Office™ documents and so on. We havefound that spatial resolution and tonal resolution are to some extentinterchangeable and that lack of spatial resolution can be compensatedby improved tonal resolution. The use of multiple gradation levelsenhances the tonal resolution in portions of the document where the datais highly discretised, such as text, drawings, etc.

This technique requires that the recording engine has a means ofrendering more than two different levels of grey or some other color,i.e. more than 0% ink and 100% ink. However most ink-jet and bubble-jetprinting devices can only deliver fixed-size drops of ink making theprinting of multi-level pixel data without reducing the spatialresolution impossible.

Solutions to this problem have been proposed in terms of using multipleinks of the same color, but different density. We refer to the documentsWO 96/12251, EP 0 850 767 A1 and U.S. Pat. No. 5,966,507 hereinincorporated by reference. A method of surrounding black pixels by greypixels for use in a 3-level electrophotographic printer was disclosed inEP 0 082 281 A1 herein incorporated by reference.

Further developments in electrophotographic printing allow the pixelsize to be adapted. Procedures to enhance the resolution in that way,working by comparing the neighbourhood of a pixel with pre-existingtemplates, are described in U.S. Pat. No. 4,847,641, U.S. Pat. No.5,193,008 and U.S. Pat. No. 5,134,495 herein incorporated by reference.

This invention however differs from the above approaches in that itcombines a number of image processing steps, different from the onesdescribed in the above-cited references, with a technology to recordmultiple gradation levels at a single pixel, with the same ink.

OBJECTS OF THE INVENTION

It is an object of the present invention to provide a printing processin which the visual quality of the obtained image is enhanced.

It is a further object to enable the softening of jagged edges at linesor curves that are not aligned with the discrete pixel grid of therecording apparatus or printer.

SUMMARY OF THE INVENTION

The above mentioned objects are realised by a method having the specificfeatures set out in claim 1. Specific features for preferred embodimentsof the invention are set out in the dependent claims. These objects arealso realised by a system having specific features of claim 11.

The above objects are mainly released by combining:

a process for printing multiple gradation levels at a single pixel, withthe same ink and

a procedure for obtaining multi-level gradation data from an abstractpage description of the document to be printed.

An image as referred to hereinafter can be anything on a document suchas text, graphics, images, etc. . . . The page data describes the layoutand content of the page to be reproduced, this data can be supplied invarious formats.

Rendering an image is the process of reproducing the image or documenton a medium or on a display screen.

A bitmap is typically a two-dimensional array of pixels. Each pixelrepresents a small square or rectangular portion of an image. In greyimages, each pixel may be represented by one value e.g. in the range of0-255. Such a bitmap is called a multi-level bitmap as every pixel inthe bitmap may take one of several available density values or gradationlevels.

In color images, each pixel is typically represented by three or morecolor components. For each color component of each pixel a value isrequired. In a system with three color components, where each colorpixel value is represented by 8 bits, each color pixel may take 256different values for each component. Consequently each pixel may take256³=16,777,216 possible values. Besides the three color values, extrabitmap values can be calculated for the black color component. Theoutput bitmap is a bitmap which is sent to the printer for rendering ofthe image by the printer.

The resolution of an image is the number of pixels used to represent alinear size of the image. For a physical representation of an image thisis expressed as the number of pixels or dots per unit of length, e.g. 16dots/mm or 400 dots/inch(dpi). For electronic image data, the resolutioncan be defined as the physical resolution if the image would be renderedon a certain size.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows various steps in a printing process according to theinvention.

FIG. 2 shows the subdivision of a high-resolution bitmap 21 into blocks.

FIG. 3 illustrates multi-level error diffusion.

FIG. 4 shows how a screening function S (k, l) is superposed on thecontinuous tone pixel L (k, l) data and is thresholded to get bi-leveloutput B (k, l).

FIG. 5 shows multi-level screening using one screening function S (k,l).

FIG. 6 shows how three-level screening with two screening functions canbe done.

FIG. 7a shows the text printed without making use of the presentinvention.

FIG. 7b shows the text printed at higher resolution.

FIG. 7c shows the text after an anti-aliasing procedure according to theinvention.

DETAILED DESCRIPTION OF THE INVENTION

According to FIG. 1 the page description 20 to be printed is processedby a Raster Image Processor (RIP) to produce high-resolution multi-levelbitmap data 21 at a horizontal resolution Hres1, a vertical resolutionVres1, and a number of gradation levels Nlev.

Then a resolution-lowering (RL) procedure converts the high-resolutionmulti-level bitmap data 21 to low-resolution multi-level bitmap data 22from horizontal resolution Hres1 to Hres2 and from vertical resolutionVres1 to Vres2, where Hres1≧Hres2 and Vres1≧Vres2 andHres2×Vres2<Hres1×Vres1. Hres2 and Vres2 correspond preferably to thehorizontal and vertical spatial resolution of the printer by which thedocument will be printed.

An image-processing (IP) step reduces the number of gradation levelsfrom Nlev to Nlevp thereby converting the low resolution multi-levelbitmap 22 to a low-gradation low-resolution output bitmap 23. Nlevp ispreferebly the number of levels that can be printed with one single inkon a single pixel or microdot.

A microdot is the smallest addressable unit on the substrate to beprinted by the output device or printer. The multi-level output bitmap23 is then sent to the printer for printing.

Most documents to be printed contain a mixture of text, graphics andimages and are given in the form of a page description 20. The pagedescription 20 is device independent. To print the document the pagedescription 20 has to be converted to a device dependent bitmap 23. Therequired number of pixels in the output bitmap 23 depends on the outputresolution of the output device. The required number of levels for eachpixel in that output bitmap 23 depends on the tone level capability ofthe output device. Such an output bitmap is preferably provided for eachcolor used to print out the document.

The first step of a printing procedure according to the currentinvention is carried out by a RIP that transforms the page description20 into a high-resolution multi-level bitmap 21 or rasterised data.

This is done e.g. by Adobe's Configurable PostScript Interpreter (CPSI).Horizontal and vertical resolution Hres1 and Vres1 can be specified tothe RIP as well as the number of gradation levels Nlev to be used.Typically we take Nlev=256. The horizontal and vertical size Hsize andVsize are also specified and determine the number of pixels Hnpix1 andVnpix1 in horizontal and vertical direction.

The spatial resolution Hres and Vres for a continuous tone image may beexpressed in pixels per millimeter. The spatial resolution Hres and Vresfor a printing device is also expressed in pixels per millimeter, orsometimes (micro)dots per inch. A typical value for the spatialresolution of an inkjet printer is Hres2=Vres2=300 dpi (dots per inch)which corresponds to about 12 microdots per mm. This means that theinkjet device can address per linear millimeter twelve microdots. Thusper square mm (mm²) a total of 144 microdots can be addressedindividually. According to a preferred embodiment of the invention, thespatial resolution of the high-resolution bitmap 21 is eight timeshigher than that of the printing device. As such, Hres1 may take thevalue of 96 microdots per mm.

If the size of the printed image is known, e.g. the horizontal sizeHsize=12 mm and the vertical size Vsize=20 mm, then also the number ofpixels Hnpix1 and lines Vnpix1 or microdots required to represent theimage at a resolution of Hres1, Vres1 by a matrix may be computed by theequations:

Hnpix1=Hsize×Hres1, Vnpix1=Vsize×Vres1.

The Nlev different gradation levels are preferably represented in theinterval [0.0, 1.0] by the numbers 0.0, 1.0/(Nlev−1), 2.0/(Nlev−1), . .. , 1.0.

The second step is a resolution-lowering procedure RL. It converts thehigh-resolution bitmap data 21 from a first spatial resolution (Hres1,Vres1) and tonal resolution Nlev to a low-resolution bitmap 22 withsecond spatial resolution (Hres2, Vres2), the same tonal resolution Nlevand the same spatial dimensions Hsize₂=Hsize₁=Hsize andVsize₂=Vsize₁=Vsize preferably in the following way:

Let Hf=Hres1/Hres2 and Vf=Vres1/Vres2 be integer numbers.

Hf is the horizontal reduction factor and Vf is the vertical reductionfactor.

Let Hnpix2=Hsize×Hres2, Vnpix2=Vsize×Vres2.

As shown in FIG. 2 to achieve the reduction, Hf×Vf pixels in thehigh-resolution (hi-res) bitmap 21 are mapped onto a single multi-levelmicrodot. When converting, the hi-res bitmap 21 may be grouped in blocksof Hf×Vf pixels. Each block is mapped onto a pixel in the low-resolution(lo-res) bitmap 22 by the RL procedure. Preferably Hf and Vf areselected such that Hf×Vf=Nlevp−1. Nlevp is preferably the number ofgradation levels of the output device. This has the following advantage:

When using reduction factors Hf and Vf, generally a rectangular areacontaining Hf×Vf pixels of the hi-res bitmap 21 is transformed to onepixel in the lo-res bitmap 22. The original pixels of a hi-res textnormally have only minimum or maximum value, i.e. the pixel is blank orit is filled “full density”. Within the hi-res rectangular area thenumber of pixels which can be filled, is within the range 0, 1, 2, . . ., Hf×Vf. As such Hf×Vf+1 fill values or “density” values are possiblefor the rectangular area.

When Hf×Vf=Nlevp−1 the fill number of the pixels within the area cansimple be converted into a density level of the lo-res pixelrepresenting the rectangular area.

Preferably the reduction factors are chosen as to comply the conditionsset by the above equation. The number of achieved density levels thusdetermines the used reduction factors.

A few possible combinations for Nlevp, Hf and Vf that achieve the aboveequation are listed in the table 1 below.

TABLE 1 Choices for Hf and Vf combining with to specific Nlevp values.Nlevp Hf Vf 3 2 1 1 2 4 3 1 1 3 5 4 1 1 4 2 2 6 5 1 1 5 7 6 1 3 2 2 3 16 8 7 1 1 7 16 15 1 5 3 3 5 1 15

Indicating the pixels in the hi-res bitmap 21 by p(i, j), with0≦i≦Hnpix1−1 and 0≦j≦Vnpixl−1, pixels p(i, j) for which

(i div Hf)=k

 (j div Vf)=l

for some integer numbers 0≦k≦Hnpix2−1 and 0≦l≦Vnpix2−1 are grouped inone block and will be mapped onto the pixel p(k, l) in the lo-res bitmap22. The operation (a div b) is the result of integer division of a by b.

The block of pixels from the hi-res bitmap 21 corresponding to a pixelp(k, l) in the lo-res bitmap 22 is denoted by B(p). It is convenient toidentify the pixel p(k, l) in the lo-res bitmap with the centre of itsblock B(p) in the hi-res bitmap. The gradation level for the pixel p(k,l) in the lo-res bitmap 22 is calculated from the pixel values in itscorresponding block B(p). Preferably the gradation level Lp(k, l) forthe pixel p(k, l) is taken equal to the average of the gradation levelsof the pixels in B(p) or equal to the gradation level closest to thismean value.

In this way the hi-res bitmap 21 is converted to the lo-res bitmap 22,which has lower spatial resolution. When the document consists e.g. outof pure black text fonts, the hi-res bitmap 21 will only contain levels0 and 1, but the lo-res bitmap 22 will contain also other grey levels inthe range 0.0, 1.0/(Nlev−1), 2.0/(Nlev−1), . . . , 1.0.

The font edges will appear softer after this operation.

It is clear that this operation combines local averaging andsubsampling, and that the local averaging operation can be replaced byother filtering methods, linear or non-linear. This is especiallyadvantageous if the hi-res bitmap 21 contains continuous tone datavalues, i.e. values within the interval [0.0, 1.0]. Alternatively we maywant to use a linear filter, e.g. a gaussian filter and allow the blocksto overlap. Let p represent the location 30 of the lo-res pixel (thecentre of B(p)) and q represent the location of a hi-res pixel in blockB(p). W represents a weight function. By linear filtering or convolutionis meant a mathematical operation that produces a weighted local averagevalue {overscore (L)} of the pixel values Lq in the hi-res block B(p)i.e.:$\overset{\_}{L} = \left( {\sum\limits_{q\quad {in}\quad {B{(p)}}}{{W\left( {q - p} \right)}L_{q}}} \right)$

Preferably for the weights W(q−p), 0≦W(q−p)≦1.0 and Σ W(q−p)=1.0 butW(q−p) may also take negative values. W is called the filter orconvolution mask.

A gaussian filter is a linear filter for which the weights Wq havegaussian values e.g. W(q−p)=t.exp (−q−p²/s), with t and s scalingfactors, q−p is equal to the distance between p and q.

Instead of taking a linear function to obtain {overscore (L)}, anon-linear function is also conceivable. An example of such a non-linearfunction is median filtering of the values of the block. More detailedinformation can be found in “A simplified approach to image processing,Randy Crane, Prentice Hall, New Jersey (1997)”.

In fact, these operations are all special cases of the more generalprocedure of resampling.

According to the sampling theory, to regain a continuous signal from itssampled version, it is advantageous to pass this signal through an ideallow-pass filter, to reconstruct the continuous signal exactly when thiscontinuous signal is band-limited in frequency space. Since images arealways limited in spatial extent, they can never really be band-limitedin frequency space, but low-pass filtering nevertheless provides anapproximation of the continuous signal. To resample a signal, theappropriate strategy is to first try to approximate the continuoussignal and then sample this continuous signal.

The RL process described herein above amounts thus to low-pass filteringfollowed by sampling.

When the operation is viewed in this way, there is no need to restrictthe method according to the current invention to the case whereHf=Hres1/Hres2 and Vf=Vres1/Vres2 are integer numbers and Hf×Vf=Nlevp−1.

The sampling operation can be interpreted as a geometric shrinking ofthe image wherein a pixel (i, j) is mapped onto (k, l)=(i/Hf, j/Vf). Butin general k and l are not integer values. To determine the pixel valuefor a pixel(k, l) of the sampled image interpolation between the pixelslying close to (Hf*k, Vf*l) in the hi-res image is advantageous. To thisend nearest neighbour interpolation, bilinear interpolation, bicubicinterpolation, etc. may be used. These techniques are described in thebook by “Crane” referred to before.

A possible draw-back of this procedure is the following. Mostconversions from abstract page data to bitmap characters use so-calledfont hinting. This font hinting guarantees that vertical or horizontallines within the same character and among different characters are allequally thick, e.g. all three stems in the letter ‘m’ are equally thick,and they are equally thick as the stems of the letter ‘n’. Theconservation of this property is not guaranteed by the above procedure.In practice however, this seems to be not of major importance, and thebenefits of the current procedure outweigh this possible drawback.

Moreover, there is the possibility in the algorithm to choose Hf and Vf(see table 1). For Nlevp=4, one can choose Hf=3, Vf=1 or Hf=1, Vf=3. Thefirst option will guarantee the vertical stems to remain balanced, whilethe latter option keeps the horizontal stems, e.g. in the letter ‘E’,balanced. Alternatively, the chosen low-pass filter involved in theabove described resampling procedure can be larger in the horizontaldirection than in the vertical direction or vice versa.

The above resampling operation can be followed by an additionalnon-linear mapping from 0.0, 1.0/(Nlev−1), 2.0/(Nlev−1), . . . , 1.0 to0.0, 1.0/(Nlev−1), 2.0/(Nlev−1), . . . , 1.0. This mapping cancompensate for the fact that the multiple levels, which the printer iscapable to produce, are not necessarily linearly related in luminance.For color documents, it is not necessary to treat all bitmaps of thecolor separation in the same way. The above outlined procedure can e.g.be applied to the bitmaps in the color separation using different blocksizes and number of levels for different bitmaps, only to the blackcomponent, etc.

A third step in the printing process according to an embodiment of thecurrent invention is an image-processing step that reduces the number ofgradation levels from Nlev to Nlevp and which converts (IP) the lo-reshigh-gradation multi-level bitmap 22 into a lo-res output bitmap 23.Nlevp is preferably the number of levels that can be printed with oneink on a single pixel or microdot. For this conversion (IP) one can usea multi-level screening procedure or multi-level error diffusion.

An example of multi-level error diffusion is multi-level Floyd-Steinbergerror diffusion, an extension of the algorithm by Floyd and Steinberg,see FLOYD R W and STEINBERG L, An adaptive algorithm for spatial greyscale, Society for Information Display 1975 Symposium Digest ofTechnical Papers, 36-37 (1975) herein incorporated by reference. In thisalgorithm the image pixels are processed in a sequential way. As shownin FIG. 3 a pixel value 25 in the range 0.0, 1.0/(Nlev−1), 2.0/(Nlev−1),. . . , 1.0 is set to its nearest value for the output 26 in the range0.0, 1.0/(Nlevp−1), 2.0/(Nlevp−1), . . . , 1.0 by quantizer 27. Theerror made by this quantisation is calculated by the error calculationcircuit 28 and is distributed over its neighbours 29, 30, 31, 32 thathave not yet been processed.

An alternative way to reduce the number of gradation levels is the useof binary or multi-level screening techniques instead of errordiffusion.

As shown in FIG. 4 screening may be performed by defining a (preferablyperiodic) screen function S(k, l) and adding this function to the pixeldata level L(k, l) that is to be screened. The result of this additionL(k, l)+S(k, l) is then thresholded by a constant function T to yieldbinary pixel values B(k, l). When e.g. both the screen function S(k, l)and pixel data levels L(k, l) take values in [0,1], the following binaryvalue B(k, l) may be assigned to the pixel (k, l)

B(k,l)=1 if L(k,l)+S(k,l)≧1,

B(k,l)=0 if L(k,l)+S(k,l)<1.

For clarity in FIG. 4, the screening operation S(k, l) is represented inonly one dimension instead of two.

This procedure can be extended to multi-level screening in the followingway:

In FIG. 5 the situation is shown for mapping continuous tone pixel dataL(k, l) (0≦L(k, l)≦1) to 4-level data N(k, l)=0, ⅓, ⅔, 1 by using ascreen function S(k, l) (0≦S(k, l)≦1)

N(k,l)=0 if 0≦3.L(k,l)+S(k,l)<1

N(k,l)=⅓ if 1≦3.L(k,l)+S(k,l)<2

N(k,l)=⅔ if 2≦3.L(k,l)+S(k,l)<3

N(k,l)=1 if 3≦3.L(k,l)+S(k,l)≦4

More generally, suppose that continuous tone pixel data L(k, l) must bemapped to N-level data N(k, l) (0, 1/(N−1), 2/(N−1), 1) by using ascreen function S(k, l) (0≦S(k, l)≦1), N(k, l) can be calculated asfollows:

let K ∈{0,1,2, . . . , N−1},

0≦L(k,l)≦1

and

0≦S(k,l)≦1

N(k,l)=K/(N−1) if K≦(N−1)L(k,l)+S(k,l)≦K+1,

N(k,l)=1 if L(k,l)=S(k,l)=1.

This is illustrated in FIG. 5 where N=4.

Alternatively, it is possible to use N−1 screening functions S₁, . . . ,S_(N−1), chosen such that for all pixels (k, l)

S ₁(k,l)<S ₂(k,l)<. . . <S _(N−1) (k,l)

and to assign

N(k,l)=0 if L(k,l)+S ₁(k,l)<1,

N(k,l)=K/(N−1) if L(k,l)+S _(f)(k,l)≧1, only for f=1,2, . . . , K.

This procedure is illustrated in FIG. 6 for N=3 (three levels 0, ½, 1,two screening functions S₁(k, l), S₂(k, l)).

The fourth step is the printing of the output bitmap by a multi-levelreproduction device such as an ink-jet printing head. Printing headsexist today that can deliver variable drop sizes of a single ink,thereby giving the opportunity to create multiple grey or color levels.The XaarJet 500 printheads, which are available in binary and greyscaleversions are an example of these. The printhead specification includes aprinting width of 70 mm with 500 nozzles, and a printing resolution of180 dpi (7 dots/mm) or 360 dpi (14 dots/mm).

The image processing procedures as outlined in the first three steps canbe combined with a printing step by such a printhead to print out theobtained multi-level bitmap data. This technique may also be used inelectrographic or electrophotographic printers having multi-levelcapabilities as described e.g. in EP 0 634 862.

FIG. 7a-7 c give an example of the effect of the different methods usedin the process on a text sample.

FIG. 7a depicts the text sample as it would be printed without makinguse of the present invention. The page data is converted to a lo-resbitmap and the text is merely printed by the low resolution printheadusing binary microdots. The binary dots can only have minimum andmaximum density. This gives an unfavourable result with dominantaliasing distortion.

FIG. 7b gives the image obtained after the first step of the method,i.e. the page data is converted to a hi-res, multi-level bitmap. In thisexample the hi-res bitmap has a resolution three times higher in thehorizontal and the vertical direction. The microdots still show only twolevels, i.e. the minimum and maximum density because the text in pagedata corresponds to a black text on a white background. This imageexists only in an electronic form and can not be printed out by thelow-res printhead. Although the bitmap is multi-level, the microdotsonly take the minimum and maximum density value.

In the second step the multi-level bitmap is converted to a multi-levelbitmap (Nlev levels) at the resolution of the printhead. The horizontaland vertical reduction factor is set to 3. This is not shown in afigure.

In the third step the number of levels of the multi-resolution bitmap isreduced to the number of levels (here Nlevp=10 levels) which can beprinted by the printhead. To obtain an easy conversion, the reductionfactors Hf and Vf and Nlevp fulfil the requirements of Hf×Vf=Nlevp−1(3×3=10−1).

The obtained lo-res, multi-level bitmap can be printed by the printheadresulting in the image of FIG. 7c. As can be seen an optimum use is madeof the printhead's multi-level capability in combination with theresolution of the printer. Due to this method a smoother image isobtained avoiding aliasing effects.

Those skilled in the art will appreciate that numerous modifications andvariations may be made to the above disclosed embodiments withoutdeparting from the spirit and scope of the present invention.

We claim:
 1. A method for rendering an image comprising the steps of:converting page data to a high-resolution multi-level bitmap, comprisingpixels p(i, j), said high-resolution bitmap having a vertical resolutionVres1 and a horizontal resolution Hres1; and having more than twopossible graduation levels; converting the high-resolution multi-levelbitmap to a low-resolution multi-level bitmap comprising pixels p(k, l),and having a vertical resolution Vres2 and a horizontal resolution Hres2and having more than two possible graduation levels; rendering thelow-resolution multi-level bitmap with a multi-level output device;wherein Hres1≧Hres2; Vres1≧Vres2; and Hres2×Vres2<Hres1×Vres1.
 2. Themethod according to claim 1 wherein said high-resolution multi-levelbitmap and said low-resolution multi-level bitmap have Nlev gradationlevels and wherein the step of rendering said low-resolution multi-levelbitmap further comprises the steps of: processing the low-resolutionmulti-level bitmap to reduce the number of gradation levels Nlev to anumber of gradation levels Nlevp to obtain an low-gradation,low-resolution output bitmap wherein Nlev>Nlevp; printing said outputbitmap on said output device.
 3. The method according to claim 1 whereinthe low-resolution multi-level bitmap is a resampled version of thehigh-resolution multi-level bitmap.
 4. The method according to claim 1wherein the gradation level of said pixel p(k, l) is determined by alocal average of the gradation levels of pixels p(i, j).
 5. The methodaccording to claim 1 wherein the gradation level of said pixel p(k, l)is determined by a weighted local average, using a linear filterfunction or convolution mask, of the gradation levels of pixels p(i, j).6. The method according to claim 1 wherein the gradation level of pixelp(k, l) is determined by a non-linear filter function of the gradationlevels of pixels p(i, j).
 7. The method according to claim 1 wherein themulti-level output device is an inkjet printer.
 8. A method forrendering an image comprising the steps of: converting page data to ahigh-resolution multi-level bitmap, comprising pixels p(i, j), saidhigh-resolution bitmap having a vertical resolution Vres1 and ahorizontal resolution Hres 1; converting the high-resolution multi-levelbitmap to a low-resolution multi-level bitmap, comprising pixels p(k,l), and having a vertical resolution Vres2 and a horizontal resolutionHres2, wherein said high-resolution multi-level bitmap and saidlow-resolution multi-level bitmap have Nlev gradation levels, andwherein Hres 1 Hres2, Vresl≧Vres2, and Hres2×Vres2<Hres1×Vres1;rendering the low-resolution multi-level bitmap with a multi-leveloutput device by processing the low-resolution multi-level bitmap toreduce the number of gradation levels Nlev to a number of gradationlevels Nlevp to obtain an low-gradation, low-resolution output bitmapwherein Nlev >Nlevp and printing said output bitmap on said outputdevice; defining a horizontal reduction factor Hf,${{Hf} = \frac{Hres1}{Hres2}};$

defining a vertical reduction factor Vf, ${{Vf} = \frac{Vres1}{Vres2}};$

and selecting Nlevp such that Hf×Vf=Nlevp−1.
 9. The method according toclaim 8 wherein the reduction factors Hf and Vf have integer values. 10.A method for rendering a color image from a plurality of color bitmapswherein at least one color bitmap is processed by a method comprisingthe steps of: converting page data to a high-resolution multi-levelbitmap comprising pixels p(i, j), said high-resolution bitmap having avertical resolution Vres1 and a horizontal resolution Hres1; saidhigh-resolution multi-level bitmap having more than two possiblegraduation levels; converting the high-resolution multi-level bitmap toa low-resolution multi-level bitmap comprising pixels p(k, l), saidlow-resolution bitmap having a vertical resolution Vres2 and ahorizontal resolution Hres2 and having more than two possible graduationlevels; converting the high-resolution multi-level bitmap to alow-resolution multi-level bitmap comprising pixels p(k, f), saidlow-resolution bitmap having a vertical resolution Vres2 and ahorizontal resolution Hres2 and having more than two possible gradationlevels; rendering the low-resolution multi-level bitmap with amulti-level color output device; Hres1≧Hres2; Vres1≧Vres2; andHres2×Vres2<Hres1×Vres1.
 11. The method according to claim 10 whereinsaid high-resolution bitmap and said low-resolution multi-level bitmaphave Nlev gradation levels and wherein the step of rendering saidlow-resolution multi-level bitmap further comprises the steps of:processing the low-resolution multi-level bitmap to reduce the number ofgradation levels Nlev to a number of gradation levels Nlevp to obtain ana low-gradation, low resolution output bitmap wherein Nlev >Nlevp; andprinting said output bitmap on said color output device.
 12. The methodaccording to claim 10 wherein the low-resolution multi-level bitmap is aresampled version of the high-resolution multi-level bitmap.
 13. Themethod according to claim 10 wherein the gradation level of said pixelp(k, l) is determined by a local average of the gradation levels ofpixels p(i, j).
 14. The method according to claim 10 wherein thegradation level of said pixel p(k, l) is determined by a weighted localaverage, using a linear filter function or convolution mask, of thegradation levels of pixels p(i, j).
 15. The method according to claim 10wherein the gradation level of pixel p(k, l) is determined by anon-linear filter function of the gradation levels of pixels p(i, j).16. The method according to claim 10 wherein the multi-level outputdevice is a color inkjet printer.
 17. A method for rendering a colourimage from a plurality of colour bitmaps wherein at least one colourbitmap is processed by a method comprising the step of: converting pagedata to a high-resolution, multi-level bitmap comprising pixels p(i, j),said high-resolution bitmap having a vertical resolution Vresl and ahorizontal resolution Hres 1: converting the high-resolution multi-levelbitmap to a low-resolution multi-level bitmap comprising pixels p(k, l),said low-resolution bitmap having a vertical resolution Vres2 and ahorizontal resolution Hres2, wherein said high-resolution bitmap andsaid low-resolution multi-level bitmap have Nlev gradation levels, andwherein Hresl Hres2, Vresl Vres2, and Hres2×Vres2 <Hresl×Vresl:rendering the low-resolution multi-level bitmap with a multi-level coloroutput device by Processing the low-resolution multi-level bitmap toreduce the number of gradation levels Nlev to a number of gradationlevels Nlevp to obtain a low-gradation, low resolution outiut bitmapwherein Nlev >Nlevp and Printing said output bitmap on said color outnutdevice; defining a horizontal reduction factor Hf,${{Hf} = \frac{Hres1}{Hres2}};$

defining a vertical reduction factor Vf, ${{Vf} = \frac{Vres1}{Vres2}};$

and selecting Nlevp such that Hf×Vf=Nlevp−1.
 18. The method according toclaim 17 wherein the reduction factors Hf and Vf have integer values.19. A system for rendering an image from page data comprising: aconvertor for converting said page data to a high resolution multi-levelbitmap comprising pixels p(i, j) and said high-resolution bitmap havinga vertical resolution Vres1 and a horizontal resolution Hres1, and saidhigh-resolution bitmap having more than two possible gradation levels; aconvertor for converting the high-resolution multi-level bitmap to alow-resolution multi-level bitmap comprising pixels p(k, l), and saidlow-resolution multi-level bitmap comprising pixels p(k, l) and havingmore than two possible gradation levels and said low-resolution bitmaphaving a vertical resolution Vres2, a horizontal resolution Hres2wherein Hres1≧Hres2; Vres1≧Vres2; and Hres2×Vres2<Hres1×Vres1, and amulti-level output means for rendering the low-resolution multi-levelbitmap.